To solve the equation \(2^{1/2} - 2^{1/4}d = 7^{9/16}\), we can start by simplifying each term.
-
Convert the left side:
- We know that \(2^{1/2} = \sqrt{2}\) and \(2^{1/4} = \sqrt[4]{2} = 2^{0.25}\).
-
Rewrite the equation: \[ \sqrt{2} - 2^{1/4}d = 7^{9/16} \]
-
To isolate \(d\), we can rearrange the equation: \[ 2^{1/4}d = \sqrt{2} - 7^{9/16} \] \[ d = \frac{\sqrt{2} - 7^{9/16}}{2^{1/4}} \]
-
Calculate the terms:
-
Calculate \( \sqrt{2} \): \(\sqrt{2} \approx 1.414\)
-
Evaluate \(7^{9/16}\): First, \(7^{9/16} = (7^{9})^{1/16}\). Calculate \(7^9\): \[ 7^9 = 40353607 \quad \text{(it may be easier to compute this using a calculator)} \] Then take \(7^{9/16} \approx 1.776\).
-
-
Substitute the values: \[ \sqrt{2} - 7^{9/16} \approx 1.414 - 1.776 \approx -0.362 \]
Now we need to divide this by \(2^{1/4}\): \(\text{Note: } 2^{1/4} \approx 1.189\).
-
Final calculation for \(d\): \[ d = \frac{-0.362}{1.189} \approx -0.304 \]
(Ensure to use a calculator for getting precise evaluations for exponents while calculating \(d\).)
Taking all above approximations into account, we can conclude: \[ d \approx -0.304 \]